A numerical study of stationary solution of viscous Burgers’ equation using wavelet

In this paper, the quadratic B-spline collocation methodis implemented to find numerical solution of theBenjamin-Bona-Mahony-Burgers (BBMB) equation. Applying theVon-Neumann stability analysis technique, we show that the method is unconditionally stable. Also the convergence of the method is proved. The method is applied on some testexamples, and numerical results have been compared with theexact solution. The numerical solutions show theefficiency of the method computationally.

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Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.454 ◽

2018 ◽

Vol 850 ◽

pp. 624-645 ◽

Cited By ~ 5

Author(s):

Brendan P. Murray ◽

Miguel D. Bustamante

Keyword(s):

Burgers Equation ◽

Numerical Study ◽

Physical Space ◽

Space Velocity ◽

Inertial Range ◽

Kuramoto Model ◽

Fast Time ◽

Phase Dynamics ◽

Velocity Statistics ◽

Phase Alignment

We present a theoretical and numerical study of Fourier-space triad phase dynamics in the one-dimensional stochastically forced Burgers equation at Reynolds number $Re\approx 2.7\times 10^{4}$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of the Kuramoto model (Chemical Oscillations, Waves, and Turbulence, Springer, 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of the Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time – a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical-space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

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Fractional Moments of Solutions to Stochastic Recurrence Equations

Journal of Applied Probability ◽

10.1239/jap/1389370094 ◽

2013 ◽

Vol 50 (4) ◽

pp. 969-982 ◽

Cited By ~ 4

Author(s):

Thomas Mikosch ◽

Gennady Samorodnitsky ◽

Laleh Tafakori

Keyword(s):

Stationary Solution ◽

Numerical Study ◽

Recurrence Equation ◽

Erlang Distribution ◽

Recurrence Equations ◽

Stochastic Recurrence ◽

Fractional Moments ◽

Independent And Identically Distributed

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

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Fractional Moments of Solutions to Stochastic Recurrence Equations

Journal of Applied Probability ◽

10.1017/s0021900200013747 ◽

2013 ◽

Vol 50 (04) ◽

pp. 969-982 ◽

Cited By ~ 1

Author(s):

Thomas Mikosch ◽

Gennady Samorodnitsky ◽

Laleh Tafakori

Keyword(s):

Stationary Solution ◽

Numerical Study ◽

Recurrence Equation ◽

Erlang Distribution ◽

Recurrence Equations ◽

Stochastic Recurrence ◽

Fractional Moments ◽

Independent And Identically Distributed

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equationXt=AtXt−1+Bt,t∈Z, where ((At,Bt))t∈Zis an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p,p∈R. Special attention is given to the case whenBthas an Erlang distribution. We provide various approximations to the moments E|X0|pand show their performance in a small numerical study.

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